Quantum operations on conformal nets

On a conformal net A, one can consider collections of unital completely positive maps on each local algebra A(I), subject to natural compatibility, vacuum preserving and conformal covariance conditions. We call quantum operations on A the subset of extreme such maps. The usual automorphisms of A (the vacuum preserving invertible unital *-algebra morphisms) are examples of quantum operations, and we show that the fixed point subnet of A under all quantum operations is the Virasoro net generated by the stress-energy tensor of A. Furthermore, we show that every irreducible conformal subnet B & SUB; A is the fixed points under a subset of quantum operations. When B & SUB; A is discrete (or with finite Jones index), we show that the set of quantum operations on A that leave B elementwise fixed has naturally the structure of a compact (or finite) hypergroup, thus extending some results of [M. Bischoff, Generalized orbifold construction for conformal nets, Rev. Math. Phys. 29 (2017) 1750002]. Under the same assumptions, we provide a Galois correspondence between intermediate conformal nets and closed subhypergroups. In particular, we show that intermediate conformal nets are in one-to-one correspondence with intermediate subfactors, extending a result of Longo in the finite index/completely rational conformal net setting